Generalized Residual Closure: General Learning Dynamics for Stability-Plasticity Compatibility

Dongxu Li, Yinuo Zhang, Hongyu Zhang, Feng Tian

Learning must acquire new capabilities while preserving both prior responsibilities and the capacity to learn again. We introduce Generalized Residual Closure (GRC), a framework for learning as recursive closure of future-relevant discrepancies: closing a residual establishes the conditions for subsequent prediction, interaction, and learning. Under a complete representation-relation description at a fixed learner-world boundary, persistent internal learning has two primitive modes: Transformation within a representation and revision of the Representation itself. We establish a local tangent decomposition under regularity assumptions and a criterion for when representation revision is necessary. In an affine model, we derive a necessary-and-sufficient condition for stability-plasticity compatibility and the unique solution of a constrained quadratic update problem, which preserves registered old responsibilities while reducing residuals with an effective safe response. We prove that reconstructive semantic protection weakly enlarges the safe-response operator relative to preserving an exact historical realization. Dynamic sufficiency and future-closure viability extend…